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May  2013, 12(3): 1487-1499. doi: 10.3934/cpaa.2013.12.1487

Asymptotic analysis of continuous opinion dynamics models under bounded confidence

1. 

Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy, Italy

Received  May 2012 Revised  July 2012 Published  September 2012

This paper deals with the asymptotic behavior of mathematical models for opinion dynamics under bounded confidence of Deffuant-Weisbuch type. Focusing on the Cauchy Problem related to compromise models with homogeneous bound of confidence, a general well-posedness result is provided and a systematic study of the asymptotic behavior in time of the solution is developed. More in detail, we prove a theorem that establishes the weak convergence of the solution to a sum of Dirac masses and characterizes the concentration points for different values of the model parameters. Analytical results are illustrated by means of numerical simulations.
Citation: Domenica Borra, Tommaso Lorenzi. Asymptotic analysis of continuous opinion dynamics models under bounded confidence. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1487-1499. doi: 10.3934/cpaa.2013.12.1487
References:
[1]

G. Aletti, G. Naldi and G. Toscani, First-order continuous models of opinion formation, SIAM J. Appl. Math., 67 (2007), 837-853. doi: 10.1137/060658679.

[2]

A. Baldassarri, U. Marini Bettolo Marconi and A. Puglisi, Kinetic models of inelastic gases, Mat. Mod. Meth. Appl. Sci., 12 (2002), 965-983. doi: 10.1142/S0218202502001982.

[3]

E. Ben-Naim, P. L. Krapivsky and S. Redner, Bifurcation and patterns in compromise processes, Phys. D, 183 (2003), 190-204. doi: 10.1016/S0167-2789(03)00171-4.

[4]

M. L. Bertotti, On a class of dynamical systems with emerging cluster structure, Jour. Diff. Eq., 249 (2010), 2757-2770. doi: 10.1016/j.jde.2010.03.014.

[5]

V. Blondel, J. M. Hendrickx and J. N. Tsitsiklis, On the $2R$ conjecture for multiagent systems, Proc. Europ. Control Conf., Kos, Greece, (2007), 874-881.

[6]

L. Boudin and F. Salvarani, Modelling opinion formation by means of kinetic equations, in "Mathematical Modeling of Collective Behavior in Socio-economic and Life Sciences" (G. Naldi, L. Pareschi and G. Toscani eds.), (2010), 245-270, Birkhauser, Boston. doi: 10.1007/978-0-8176-4946-3_10.

[7]

L. Boudin and F. Salvarani, The quasi-invariant limit for a kinetic model of sociological collective behavior, Kinet. Relat. Models, 2 (2009) 433-449. doi: 10.3934/krm.2009.2.433.

[8]

L. Boudin and F. Salvarani, A kinetic approach to the study of opinion formation, M2AN Math. Model. Numer. Anal., 43 (2009), 507-522. doi: 10.1051/m2an/2009004.

[9]

C. Canuto, F. Fagnani and P. Tilli, A Eulerian approach to the analysis of rendez-vous algorithms, Proc. of 2008 IFAC Conf., (2008), 9039-9044.

[10]

G. Como and F. Fagnani, Scaling limits for continuous opinion dynamics systems, Ann. Appl. Probab., 21 (2011), 1537-1567. doi: 10.1214/10-AAP739.

[11]

S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, Journal of Statistical Physics, 120 (2005), 253-277. doi: 10.1007/s10955-005-5456-0.

[12]

G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents, Adv. Comp. Sys., 3 (2001), 87-98. doi: 10.1142/S0219525900000078.

[13]

B. Düring, P. Markowich, J-F. Pietschmann and M-T.Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proc. R. Soc. Lond. A, 465 (2009)(2112), 3687-3708. doi: 10.1098/rspa.2009.0239.

[14]

S. Galam, A review of Galam models, Int. J. Mod. Phys. C, 409 (2008), 3687-3708.

[15]

U. Krause, A discrete nonlinear and non-autonomous model of consensus formation, in "Communications in Difference Equations" (S. Elaydi, G. Ladas, J. Popenda and J. Rakowski eds.), Gordon and Breach Science Publ., Amsterdam, (2000) 227-236.

[16]

J. Lorenz, A stabilization theorem for continuous opinion dynamics, Phys. A, 355 (2005), 217-223. doi: 10.1016/j.physa.2005.02.086.

[17]

J. Lorenz, Continuous opinion dynamics under bounded confidence: A survey, Internat. J. Modern Phys. C, 18 (2007), 1819-1838. doi: 10.1142/S0129183107011789.

[18]

S. McNamara and W. R. Young, Kinetics of a one dimensional granular medium in the quasi elastic limit, Phys. Fluids A, 5 (1993), 34-45. doi: 10.1063/1.858896.

[19]

L. Pareschi and G. Toscani, Self-similarity and power-like tails in nonconservative kinetic models, J. Stat. Phys., 124 (2006), 747-779. doi: 10.1007/s10955-006-9025-y.

[20]

G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.

[21]

G. Weisbuch, G. Deffuant, F. Amblard and J. P. Nadal, Meet, discuss, and segregate!, Complexity, 7 (2002), 55-63. doi: 10.1002/cplx.10031.

show all references

References:
[1]

G. Aletti, G. Naldi and G. Toscani, First-order continuous models of opinion formation, SIAM J. Appl. Math., 67 (2007), 837-853. doi: 10.1137/060658679.

[2]

A. Baldassarri, U. Marini Bettolo Marconi and A. Puglisi, Kinetic models of inelastic gases, Mat. Mod. Meth. Appl. Sci., 12 (2002), 965-983. doi: 10.1142/S0218202502001982.

[3]

E. Ben-Naim, P. L. Krapivsky and S. Redner, Bifurcation and patterns in compromise processes, Phys. D, 183 (2003), 190-204. doi: 10.1016/S0167-2789(03)00171-4.

[4]

M. L. Bertotti, On a class of dynamical systems with emerging cluster structure, Jour. Diff. Eq., 249 (2010), 2757-2770. doi: 10.1016/j.jde.2010.03.014.

[5]

V. Blondel, J. M. Hendrickx and J. N. Tsitsiklis, On the $2R$ conjecture for multiagent systems, Proc. Europ. Control Conf., Kos, Greece, (2007), 874-881.

[6]

L. Boudin and F. Salvarani, Modelling opinion formation by means of kinetic equations, in "Mathematical Modeling of Collective Behavior in Socio-economic and Life Sciences" (G. Naldi, L. Pareschi and G. Toscani eds.), (2010), 245-270, Birkhauser, Boston. doi: 10.1007/978-0-8176-4946-3_10.

[7]

L. Boudin and F. Salvarani, The quasi-invariant limit for a kinetic model of sociological collective behavior, Kinet. Relat. Models, 2 (2009) 433-449. doi: 10.3934/krm.2009.2.433.

[8]

L. Boudin and F. Salvarani, A kinetic approach to the study of opinion formation, M2AN Math. Model. Numer. Anal., 43 (2009), 507-522. doi: 10.1051/m2an/2009004.

[9]

C. Canuto, F. Fagnani and P. Tilli, A Eulerian approach to the analysis of rendez-vous algorithms, Proc. of 2008 IFAC Conf., (2008), 9039-9044.

[10]

G. Como and F. Fagnani, Scaling limits for continuous opinion dynamics systems, Ann. Appl. Probab., 21 (2011), 1537-1567. doi: 10.1214/10-AAP739.

[11]

S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, Journal of Statistical Physics, 120 (2005), 253-277. doi: 10.1007/s10955-005-5456-0.

[12]

G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents, Adv. Comp. Sys., 3 (2001), 87-98. doi: 10.1142/S0219525900000078.

[13]

B. Düring, P. Markowich, J-F. Pietschmann and M-T.Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proc. R. Soc. Lond. A, 465 (2009)(2112), 3687-3708. doi: 10.1098/rspa.2009.0239.

[14]

S. Galam, A review of Galam models, Int. J. Mod. Phys. C, 409 (2008), 3687-3708.

[15]

U. Krause, A discrete nonlinear and non-autonomous model of consensus formation, in "Communications in Difference Equations" (S. Elaydi, G. Ladas, J. Popenda and J. Rakowski eds.), Gordon and Breach Science Publ., Amsterdam, (2000) 227-236.

[16]

J. Lorenz, A stabilization theorem for continuous opinion dynamics, Phys. A, 355 (2005), 217-223. doi: 10.1016/j.physa.2005.02.086.

[17]

J. Lorenz, Continuous opinion dynamics under bounded confidence: A survey, Internat. J. Modern Phys. C, 18 (2007), 1819-1838. doi: 10.1142/S0129183107011789.

[18]

S. McNamara and W. R. Young, Kinetics of a one dimensional granular medium in the quasi elastic limit, Phys. Fluids A, 5 (1993), 34-45. doi: 10.1063/1.858896.

[19]

L. Pareschi and G. Toscani, Self-similarity and power-like tails in nonconservative kinetic models, J. Stat. Phys., 124 (2006), 747-779. doi: 10.1007/s10955-006-9025-y.

[20]

G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.

[21]

G. Weisbuch, G. Deffuant, F. Amblard and J. P. Nadal, Meet, discuss, and segregate!, Complexity, 7 (2002), 55-63. doi: 10.1002/cplx.10031.

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